The string effective action at tree level contains, in its bosonic sector, the Einstein-Hilbert term, the dilaton, and the axion, besides scalar and gauge fields coming from the Ramond-Ramond sector. The reduction to four dimensions brings to scene moduli fields. We generalize this effective action by introducing two arbitrary parameters, $\ensuremath{\omega}$ and $m$, connected with the dilaton and axion couplings. In this way, more general frameworks can be analyzed. Regular solutions with a bounce can be obtained for a range of (negative) values of the parameter $\ensuremath{\omega}$ which, however, exclude the pure string configuration ($\ensuremath{\omega}=\ensuremath{-}1$). We study the evolution of scalar perturbations in such cosmological scenarios. The predicted primordial power spectrum decreases with the wave number with spectral index ${n}_{s}=\ensuremath{-}2$, in contradiction with the results of the Wilkinson microwave anisotropy probe (WMAP). Hence, all such effective string motivated cosmological bouncing models seem to be ruled out, at least at the tree level approximation.